Table Of ContentPossible Anderson transition below two dimensions in disordered systems of
noninteracting electrons
Yoichi Asada1, Keith Slevin2, and Tomi Ohtsuki3,4
1Department of Physics, Tokyo Institute of Technology,
2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan
2Department of Physics, Graduate School of Science,
6
Osaka University, 1-1 Machikaneyama,
0
Toyonaka, Osaka 560-0043, Japan
0
3Department of Physics, Sophia University,
2
Kioi-cho 7-1, Chiyoda-ku, Tokyo 102-8554, Japan
n 4 CREST, JST
a
(Dated: February 2, 2008)
J
2 We investigate the possibility of an Anderson transition below two dimensions in disordered
1 systems of non-interacting electrons with symplectic symmetry. Numerical analysis of energy level
statistics and conductance statistics on Sierpinski carpets with spin-orbit coupling indicates the
] occurrence of an Anderson transition below two dimensions.
n
n
- Accordingtothe scalingtheoryofAbrahamset al. for 1.5
s
i non-interactingelectronsindisorderedsystems,allstates
d 3D
. are localized in two dimensions (2D) and the Anderson 1.0
t
a transition occurs only above 2D [1]. This prediction ap-
m plies to systems with orthogonalsymmetry, i.e., systems 0.5 lnΛ3Dsymp
Λ) c
- with time reversal and spin rotation symmetry [2, 3, 4].
nd Accordingly, it is believed that the lower critical dimen- β(ln 0.0 2D
sion for the Anderson transition in systems with orthog-
o
c onal symmetry is -0.5 lnΛ2Dsymp
[ c
d(orth) =2. (1)
4 L
-1.0
v The prediction of Abrahams et al. does not apply to
5 -5 -4 -3 -2 -1 0 1 2 3 4 5
1 systems with symplectic symmetry, i.e. in systems with lnΛ
3 time reversal symmetry but in which spin rotation sym-
8 metry is broken by spin-orbit coupling [2, 5]. For such
FIG. 1: The numerically estimated β functions for systems
0 systems it is known that there is a transition in 2D, if
with symplectic symmetry in 2D and 3D (solid lines) [6, 7]
5
the spin-orbit interaction is sufficiently strong. as well as the schematic ones for systems with orthogonal
0
The prediction of Abrahams et al. rests on an argu- symmetry (dotted lines).
/
t mentconcerningtheformoftheβ functionthatdescribes
a
m the scaling of the conductance. Reasonable assumptions
concerning the asymptotic behavior of this function in transitionandthispropertyshouldbecommontoallthe
-
d the strongly metallic and localized limits, respectively, β functions. Also the slope at the zero, which is related
n andtheassumptionthattheβ functionismonotoniclead to the critical exponent ν describing the divergence of
o
to their conclusion. As we explain below it is this latter the localization length, should be the same for all the β
c
: assumption that does not hold in systems with symplec- functions.
v
tic symmetry. The solid lines in Fig. 1 are numerical estimates of
i
X In Fig. 1 we show the β function that describes the theβ function(2)forsystemswithsymplecticsymmetry
r scaling of the quantity Λ[3] [6, 7]. The β function tends to d−2, with d being the
a
dimensionality, in the metallic limit and becomes nega-
dlnΛ
β(lnΛ)= . (2) tive in the localized limit. In contrast to systems with
dlnL
orthogonal symmetry (shown schematically, see Ref. [3]
Here Λ is the ratio of the quasi-1Dlocalizationlength to for numerical result) the β functions are non-monotonic
thesystemwidthforelectronsonalongquasi-1Dsystem andafixedpointexistsevenin2D.Itisthoughtthatthe
of width L. The β functions that describe the scaling of β function for the conductance behaves similarly [8, 9].
different physical quantities are expected to differ in de- Lookingat Fig. 1 the obvious questionis, “What hap-
tail, but for systems with the same dimensionality and pens below 2D for systems with symplectic symmetry?”
symmetry all are expected to share some common prop- Doestheβ functionchangediscontinuouslyandsuddenly
erties. For example, a zero of the β function indicates a become monotonic? Or does it change smoothly? In the
2
latter case, we expect the Anderson transition to per- between lattice sites with the hopping matrices R(i,j)
sist below 2D. This is a long standing problem[10]. Our sampled uniformly from the group SU(2). Apart from
numerical results suggest that the Anderson transition the fractal lattice the model is exactly as described in
persists below 2D for systems with symplectic symme- [6, 17] and the reader should refer to these for further
try. It is known that states in (quasi-)1D systems with details.
symplectic symmetry are always localized, so the lower To investigate the localization of electrons on these
criticaldimensionmustbeatleastone. Thiswouldmean fractals we have examined the energy level statistics of
that (4). Inparticular,wehavelookedatthedistributionP(s)
ofthe nearestneighborlevelspacings measuredin units
1≤d(symp) <2. (3)
L of the mean level spacing [18]. It is known that P(s) is
well approximated by the Wigner surmise for the Gaus-
To study this problem we have simulated non-
siansymplecticensembleP (s)=As4e−Bs2 inthe ex-
interacting electrons on the Sierpinski carpet [11] with GSE
tended limit, and the Poisson distribution P (s) = e−s
spin-orbitcoupling. The Sierpinskicarpetis constructed P
in the localized limit. We have calculated the energy
byrepeatingthefollowingiteration. Westartfromab×b
eigenvaluesusingeitherLAPACKortheLanczosmethod
square lattice with a central c×c square removed. This
[19]. Anunfoldingprocedurehasbeenappliedtothelevel
first generation or “generator” is denoted as SC(b,c,1).
spacings to compensate for the variation with energy of
At each step the next generation is constructed by mag-
theensembleaveragedensityofstates. Foraquantitative
nifying the lattice b times and replacing each site of the
analysis of P(s), we define
currentgenerationwith the generator. The linear size of
the kth generation SC(b,c,k) is L = bk and the number s0dsP(s)− s0dsP (s)
of sites is N = (b2 − c2)k. The Sierpinski carpet be- Y = R0 R0 P . (5)
comes a true fractalin the mathematicalsense when the s0 R0s0dsPGSE(s)−R0s0dsPP(s)
generation number k tends to infinity. This true frac-
In the extended limit Y = 1, and in the localized limit
tal is denoted as SC(b,c). According to a lot of stud- s0
Y =0. Note that since each energy level has a twofold
ies of thermal magnetic phase transitions, the Sierpin- s0
Kramers’ degeneracy we consider only distinct eigenval-
ski carpets are useful to study phase transitions in frac-
ues when calculating the spacing distribution.
tal dimensionality less than two [12] (see also a recent
We have simulated lattices SC(5,1,k) up to k =
work Ref. [13] and references therein). We have studied
4. Calculation of higher generations is not possible at
SC(3,1,k)andSC(5,1,k)withfixedboundaryconditions
present because the CPU time required is too large. For
imposed in both directions.
the first generation k = 1 the number of eigenvalues is
A fractal system has a fractal dimension d . This di-
f
small. Therefore, we restricted our analysis to a single
mension describes how the mass of the fractal depends
pairofconsecutivelevelspersample,thesmallestpositive
onthesizeofthe fractal. Thefractaldimensiond ofthe
f
Sierpinski carpet SC(b,c), defined by N = Ldf, is equal eigenvalueandthelargestnegativeeigenvalue. Whenwe
to d =ln(b2−c2)/lnb. We find d ≈1.893 for SC(3,1) choosethepairoflevelsinthisspecialway,theprobabil-
f f ityP˜(s)ofalevelspacingwithavaluesisequaltosP(s).
and d ≈ 1.975 for SC(5,1). A fractal system also has
f Therefore we multiply P˜(s) by 1/s to obtain level spac-
a spectral dimension d [14]. This dimension describes
s
ingdistributionP(s). We simulated1000000samplesfor
the spectra of the low energy vibrations on the fractal.
k =1. For higher generations the number of eigenvalues
Numericalstudies ofthe Andersontransitioninbifractal
is greater so we reverted to the usual procedure of ana-
systemsindicatethatthisistherelevantdimensionwhen
lyzing a sequence of consecutive levels. The number of
discussing the Anderson transition [4, 15]. The spectral
samples was reduced accordingly. We used levels in the
dimension of the Sierpinski carpet has been estimated
interval E = [−0.3,0.3]. We simulated 15000, 600 and
bysimulatingaclassicalrandomwalkonthefractal[16].
10 samples for k =2, 3 and 4, respectively. The total
ThevaluesreportedforSC(3,1)andSC(5,1)are,respec-
number of the level spacings for each of the (k,W) pair
tively, d =1.785±0.008 and d =1.940±0.009.
s s is thus of order 105 to 106.
We have simulated the SU(2) model [6, 17].
Figure 2 shows Y as a function of disorder W for
1.5
H =Xǫic†iσciσ − X R(i,j)σσ′c†iσcjσ′. (4) SC(5,1,k). There are large corrections to scaling in the
i,σ hi,ji,σ,σ′ firstgeneration. The dataforthe highergenerationsk =
2,3and4indicatethatanAndersontransitionoccurson
Here c† (c ) denotes the creation (annihilation) oper- SC(5,1) around W ≈5. When disorder is weak enough,
iσ iσ
ator of an electron at the site i with spin σ. This Y increases with k for k ≥ 2 indicating a delocalized
1.5
model has symplectic symmetry. There is an on-site phase. Whendisorderisstrongenough,Y decreasesas
1.5
random potential ǫ which is identically and indepen- k increases indicating a localized phase.
i
dently distributed with uniform probability in the range Further analysis of the level statistics on SC(5,1) is
[−W/2,W/2]. There is also nearest neighbor hopping based on the assumption that finite size scaling applies
3
TABLE I: The details of the scaling analyses for SC(5,1,k). Data with W in [4.2,5.7] have been used. The range of Y
s0
has also been restricted as indicated. Here, Np is the number of parameters, Nd the number of data, Q the goodness of fit
probability, and Yc the critical valueof Ys0. The precision of the fitted parameters is expressed as 95% confidenceintervals.
s0 Ys0 nR,nI,mR,mI Np Nd Q Wc Yc ν y
0.5 [0.60,0.93] 4,1,2,0 11 63 0.3 4.99±0.02 0.791±0.009 3.50±0.11 −0.80±0.09
1.5 [0.34,0.77] 3,1,2,0 10 63 0.7 4.98±0.03 0.536±0.013 3.44±0.15 −0.77±0.11
1.0
0.7
0.8
)
d
0.6 cte 0.6
e
Y15. orr
c
0.4 ( 5 0.5
1.
Y
k=1 (L=5)
0.2 k=2 (L=25) 0.4
k=3 (L=125)
k=4 (L=625)
0.0
3 4 5 6 7 10−10 10−8 10−6 10−4 10−2
W L/ξ
FIG. 2: Y1.5 vs. W for SC(5,1,k) (k =1,2,3,4). The solid FIG. 3: The numerical data Y1.5 after subtraction of the
linesarethebestfitsof(7)tothedatainsidethebox(shown corrections to scaling plotted as function of theratio L/ξ.
by dashed lines). Outside the box the fit deviates from the
numericaldatabecausetheexpansion parametersψL1/ν and
w are nolonger small. disorder w = (W − W)/W up to the order m and
c c R
m . The best fit to data is determined by minimizing
I
the χ2 statistics and the precision of the parameters are
to fractals in the same way as to systems with integral
determined using the Monte Carlo method [23].
dimension. Accordingly the statistic Y should, in the
s0 The best fit is shown in Fig. 2 and the results of the
absence ofany correctionsto scaling,obey the single pa-
scaling analyses for s = 0.5 and s = 1.5 are tabulated
0 0
rameter scaling law
in Table I. The estimates of the criticaldisorderW and
c
thecriticalexponentν fors =0.5ands =1.5arecon-
Y =f (L/ξ)=F ψL1/ν . (6) 0 0
s0 ± 0(cid:16) (cid:17) sistentasrequired. Ourestimateν =3.4±0.2forSC(5,1)
is clearly different from the value ν = 2.746±0.009 in
Here ξ is the localization(correlation) length, ν the crit-
2D[6], reflecting the difference of the dimensionality. To
ical exponent for the divergence of ξ, and ψ a smooth
exhibit single parameter scaling the data are re-plotted,
function of disorder W that crosses zero linearly at the
after subtraction of the corrections due to an irrelevant
critical disorder W . The subscript ± distinguishes the
c variable, in Fig. 3. The two branches correspond to the
scaling function in the delocalized and localized phases.
delocalized and localized phases.
It is, however, clear from Fig. 2 that corrections to scal-
To complement our study of the level statistics, we
ing are not negligible. Taking account of corrections up
have also analyzed statistics of the Landauer conduc-
to the first order in an irrelevant variable φLy we have tance g (in units of e2/h) [21, 22, 24]. Two perfect leads
[20, 21, 22] of width L = 5k are attached to the Sierpinski carpet
SC(5,1,k) and the recursive Green’s function method is
Ys0 =F0(cid:16)ψL1/ν(cid:17)+φLyF1(cid:16)ψL1/ν(cid:17). (7) usedtocalculatetheconductance[25]. Thehoppingma-
trices in the direction transverse to the current are set
For the purpose of fitting, the scaling functions F0 and to the unit matrix. As a scaling quantity, we choose the
F1 are approximated by a power series up to the order typical conductance
n and n in ψL1/ν. The functions ψ and φ are also
R I
approximated by expansions in terms of dimensionless g =ehlngi. (8)
typ
4
0.8
3 k=2 (L=25) symplectic
k=3 (L=125) orthogonal
k=4 (L=625)
0.6
2 ν SC(5,1)
0.4
1/
p
y
gt
0.2
1
0.0
1 2 3
d
s
0
2 3 4 5 6 7 FIG.5: 1/ν vs. ds forsystemswithorthogonalandsymplec-
W ticsymmetries. Thevaluesfor2Dand3Dsymplecticsystems
are from [6] and [7], respectively. The estimates for systems
FIG. 4: The typical conductance gtyp vs. W for SC(5,1,k) with orthogonal symmetry with dimension 2 < ds < 3 are
(k=2,3,4). The lines are a guideto the eyeonly. from [4] and that in 3D is from [20]. (The exponent ν for
fractalswith2<d <3wasalsoestimatedinRef.[15]). The
s
solid line indicates ν =1/(ds−2).
Here hi means the ensemble average. We have set the
Fermi energy to E =0 and have accumulated 3000 sam-
dotted line in Fig. 5) that the lower critical dimension is
ples for k =2,3 and from 150 to 300 samples for k = 4.
1inthiscase. However,ourestimateofthecriticalexpo-
Figure 4 shows the typical conductance g as a func-
typ nentonSC(5,1)is wellbelowthe dottedline, suggesting
tion of W for SC(5,1,k). These data also indicate the
that the lower critical dimension is closer to 2 than to 1.
occurrence of an Anderson transition at about W ≈5.
To confirm the occurrence of the Anderson transition
We have also analyzed energy level statistics and con-
below 2D for systems of non-interacting electrons in dis-
ductance statistics on SC(3,1,k) up to the fourth gen-
orderedsystemswithsymplecticsymmetry,andtodeter-
eration. We did not find any evidence for a delocalized
mine more precisely the lowercriticaldimension, further
phase. We cannotconcludefromthis thatSC(3,1)isbe-
numerical studies on a variety of fractals over a larger
lowthelowercriticaldimensionbecausetheSU(2)model
range of generations are needed. Analysis of fractals
hassignificantrandomnessevenwhentheon-siterandom
with d intermediate between the values for SC(3,1) and
potential is zero. s
SC(5,1)might be particularly helpful. The nature of the
In Fig. 5 we have plotted the available results for the
possible delocalized phase also warrants further study.
critical exponent for systems with orthogonal and sym-
Recentlyelectrontransportthroughsystemswithfrac-
plectic symmetries. For 2D and 3D system with sym-
tal perimeter was studied experimentally [27]. Electron
plectic symmetry the critical exponent are estimated to
transportthroughfractalstructurescanbeaninteresting
be ν = 2.746±0.009 in 2D [6] and ν = 1.37±0.02 in
topic for further theoretical and experimental investiga-
3D [7]. For systems with orthogonal symmetry we have
tion.
used the estimates reported in [4, 20]. In the figure we
One of the authors (Y.A.) is supported by Research
plot 1/ν as a function of d .
s Fellowships of the Japan Society for the Promotion of
We recall that the slope of the β function at the fixed
Science for Young Scientists.
point is equal to 1/ν. Assuming that the β function
changescontinuouslyasthedimensionalitychanges,then
just above the lower critical dimension the fixed point
should correspond to the maximum of the β function.
We therefore expect the critical exponent ν to diverge [1] E. Abrahams, P. W. Anderson, D. C. Licciardello, and
at the lower critical dimension. This is consistent with T. V.Ramakrishnan, Phys. Rev.Lett. 42, 673 (1979).
the theoretical analysis of the nonlinear σ model with [2] S. Hikami, A. I. Larkin, and Y. Nagaoka, Prog. Theor.
orthogonal symmetry where it was found that ν = 1/ǫ Phys. 63, 707 (1980).
[3] A. MacKinnon and B. Kramer, Z. Phys.B 53, 1(1983).
forǫ=d−2≪1[26]. Thenumericaldatafororthogonal
[4] M. Schreiber and H. Grussbach, Phys. Rev. Lett. 76,
systems shown in Fig. 5 are consistent with this.
1687 (1996).
For systems with symplectic symmetry fewer data are [5] T. Ando,Phys. Rev.B 40, 5325 (1989).
available[6,7]. Whenonlytheestimatesforν in2Dand [6] Y. Asada, K. Slevin, and T. Ohtsuki, Phys. Rev. B 70,
3D are considered, it can leave the impression (cf. the 035115 (2004).
5
[7] Y. Asada, K. Slevin, and T. Ohtsuki,J. Phys. Soc. Jpn. and H.B. Shore,Phys. Rev.B 47, 11487 (1993).
Suppl.74, 238 (2005). [19] J. K. Cullum and R. A. Willoughby, Lanczos Algo-
[8] F. Wegner, Nucl.Phys. B316, 663 (1989). rithms for Large Symmetric Eigenvalue Computations
[9] S.Hikami, Prog. Theor. Phys.Suppl. 107, 213 (1992). (Birkh¨auser, Boston, 1985).
[10] S.Hikami, J. Phys. (Paris) Lett. 46, L719 (1985). [20] K. Slevin and T. Ohtsuki, Phys. Rev. Lett. 82, 382
[11] B. B. Mandelbrot, The Fractal Geometry of Nature (1999).
(CA:Freeman, San Francisco, 1982). [21] K. Slevin, P. Markoˇs, and T. Ohtsuki, Phys. Rev. Lett.
[12] Y. Gefen, Y. Meir, B. B. Mandelbrot, and A. Aharony, 86, 3594 (2001).
Phys.Rev.Lett. 50, 145 (1983). [22] K. Slevin, P. Markoˇs, and T. Ohtsuki,Phys. Rev. B 67,
[13] P. Monceau and M. Perreau, Phys. Rev. B 63, 184420 155106 (2003).
(2001). [23] W.H.Press,S.A.Teukolsky,W.T.Vetterling,andB.P.
[14] T. Nakayama and K. Yakubo, Fractal Concepts in Con- Flannery,NumericalRecipesinFortran (CambridgeUni-
densed Matter Physics (Springer-Verlag, Berlin, 2003). versity Press, Cambridge, 1992).
[15] I. Travˇenec and P. Markoˇs, Phys. Rev. B 65, 113109 [24] B. Shapiro, Phil. Mag. B 56, 1031 (1987).
(2002). [25] T. Ando,Phys. Rev.B 44, 8017 (1991).
[16] S. Fujiwara and F. Yonezawa, Phys. Rev. E 51, 2277 [26] S. Hikami, Phys.Rev. B 24, 2671 (1981).
(1995). [27] H. A. Walling, D. P. Dougherty, D. P. Druist, E. G.
[17] Y. Asada, K. Slevin, and T. Ohtsuki, Phys. Rev. Lett. Gwinn, K. D. Maranowski, and A. C. Gossard, Phys.
89, 256601 (2002). Rev. B 70, 045312 (2004).
[18] B.I.Shklovskii,B.Shapiro,B.R.Sears,P.Lambrianides,
